Flow patterns at the major T-junctions of the dog descending aorta

Using a novel technique developed in our own laboratory, an isolated transparent arterial segment containing the whole descending aorta and its four major branches was prepared from a dog. The flow patterns at each aortic T-junction were studied in detail under the conditions of steady flow by means of flow visualization and cinemicrographic techniques. It was found that a standing recirculation zone consisting of a pair of thin-layered spiral secondary flows located symmetrically about the common median plane of the aorta and side branches was formed at each T-junction over a wide range of flow conditions including the time-averaged estimated mean values of physiological flow rates and flow rate ratios. The results support the recent in vivo findings by other investigators that flow reversal occurs at some junctions of the dog abdominal aorta during each cardiac cycle. The flow patterns at the aortic T-junctions were very much similar to those previously observed in various glass model T-junctions. However, due to the particular anatomical structure of the vessel wall at each branching site (the curvature of the wall was very sharp at the flow divider, but gently rounded at the bend opposite to it) no recirculation zone was formed in the side branches. At a given flow rate ratio, the measured critical Reynolds numbers for the formation of spiral secondary flows and fully developed disturbed flows were much higher in aortic T-junctions than those in glass model T-junctions having equivalent branching angles and diameter ratios. These results indicate that, in the circulation, conditions at arterial T-junctions appear to be optimal for minimizing the formation of disturbed flows.     

Flow in t-bifurcations: Effect of the sharpness of the flow divider

The local geometry of a bifurcation has been hypothesized to be a potential geometrical risk factor for the development of atherosclerosis. While flow division and branch area ratios clearly affect the flow field, the importance of the flow divider shape is not as clear. A fast spectral element computational fluid mechanics (CFD) solver was used to simulate flow through 90 ° T-bifurcations with three different flow divider shapes. Other factors, such as flow partition, area ratio, and bifurcation angle, were kept constant. A Reynolds number range of 15 to 350 was studied to bracket experimental results in the literature. The variation in the sharpness of the corners had a dramatic effect on both the flow field and wall shear stress distribution in the side branch, but little effect on the flow in the main tube. The magnitude of reverse velocities and wall shear stress in the side branch increased linearly over a physiological range of Reynolds number and corner shape. This paper verifies the accuracy and usefulness of spectral element CFD in studying three-dimensional hemodynamics.     

Numerical simulation of flow fields in a tube with two branches

In the present study, a numerical calculation procedure based on the finite volume method was employed to simulate flow fields in double-branched tubes. The configuration was a tube with two vertical branches; the two branches were either on the same side or on the opposite side. The study focused on the baseline flow fields and the possible flow interaction between the two branches. The branching ratio and the branch /main tube diameter ratio were fixed in this study. The results showed that when the two branches were on the same side, the low/oscillating shear regions were found on the ventral walls of the branches and on the dorsal wall of the main tube distal to the branches. The flow field proximal to each branch was similar to that in a single-branched tube when the two branches were distant. When the branches were on the opposite side with the staggering distance S=0 (symmetric case), the low/oscillating shear regions were found on the lateral walls of the main tube. As S increased, the interaction between the two branches weakened, the low/oscillating shear regions were found on the lateral walls of the main tube to the side of the second branch. The flow field near the branch was significantly different from that of a single-branched tube. Care should be taken on localization of plaques in multi-branched vessels due to the flow pattern change. The numerical results were qualitatively consistent with what observed experimentally, by other investigators.     

A numerical simulation of steady flow fields in a bypass tube.

Steady flow in a complete by-pass tube was simulated numerically. The study was to consider a complete flow field, which included both the by-pass and the host tubes. The changes of the hemodynamics were investigated with three parameters: the inlet flow Reynolds number (Re), anastomotic angle (alpha) and the position of the occlusion in the host tube. The baseline flow field was set up with Re=200, alpha=45 degrees and the centered position of occlusion. The parametric study was then conducted on combination of Re=100, 200, 400, alpha=35 degrees, 45 degrees, 60 degrees, 75 degrees, 90 degrees and three occlusion positions: left, center and right. It was found that in the baseline case, large slow/recirculation flows could be seen in the host tube both upstream and downstream of the occlusion. The separation points were on the opposite walls to the junctions. Recirculation zones were also found near the toe and in the proximal outer wall of the by-pass tube. Their sizes were about one diameter of the tube or smaller. In some cases, pairing vortices could be seen in the host tube upstream of the occlusion. The shear rate distribution associated with the flow fields was presented. The flow pattern obtained was agreeable to those observed experimentally by other investigators. The difference of the flow fields between a complete bypass and simple anastomosis was discussed. The present numerical code provides a preliminary simulation/design tool for bypass graft flows.     

Vortex generation in pulsatile flow through arterial bifurcation models including the human carotid artery

Visualization experiments were performed to elucidate the complicated flow pattern in pulsatile flow through arterial bifurcations. Human common carotid arteries, which were made transparent, and glass-models simulating Y- and T-shaped bifurcations were used. Pulsatile flow with wave forms similar to those of arterial flow was generated with a piston pump, elastic tube, airchamber, and valves controlling the outflow resistance. Helically recirculating flow with a pattern similar to that of the horseshoe vortex produced around wall-based protuberances in circular tubes was observed in pulsatile flow through all the bifurcations used in the present study. This flow type, which we shall refer to as the horseshoe vortex, has also been demonstrated to occur at the human common carotid bifurcation in steady flow with Reynolds numbers above 100. Time-varying flows also produced the horseshoe vortex mostly during the decelerating phase. Fluid particles of dye solution approaching the bifurcation apex diverged, divided into two directions perpendicularly, and then showed helical motion representing the horseshoe vortex formation. While this helical flow was produced, the stagnation points appeared on the wall upstream of the apex. Their position was dependent upon the flow distribution ratio between the branches in the individual arteries. The region affected by the horseshoe vortex was smaller during pulsatile flow than during steady flow. Lowering the Reynolds number together with the Womersley number weakened the intensity of helical flow. A separation bubble, resulting from the divergence or wall roughness, was observed at the outer or inner wall of the branch vessels and made the flow more complicated.     

An experimental analysis of the flow field in a three-dimensional model of the human carotid artery bifurcation

Steady flow measurements were carried out in a rigid three-dimensional model of the human carotid artery bifurcation at a Reynolds number of 640 and a flow division ratio of 50/50. Both axial and secondary velocities were measured with a laser-Doppler anemometer. In the bulb opposite to the flow divider a zone with negative axial velocities was found with a maximal diameter of about 60% of the local diameter of the branch and a cross-sectional extent of about 25% of the local cross-sectional area. In the bulb the maximum axial velocity shifted towards the divider wall and at the end of the bulb an axial velocity plateau arose near the non-divider wall. Halfway through the bulb, secondary flow showed a vortex through which fluid flowed towards the divider wall near the bifurcation plane and back towards the non-divider wall near the upper walls.     

Erythrocyte sedimentation rate II. Effects of tilt angle in saline solution

We have measured the sedimentation curves of swine erythrocytes in a physiological saline solution in inclined glass tubes. The curves are well fitted to the exponential type equation l = a[1 - exp(-bt)] for the tilt angle theta in the range of theta less than 80 degrees and hematocrits from 10 to 50%, where l and t are the medium length along the tube and the elapsed time from the sample injection, respectively. The coefficient a increases with theta and b is proportional to sin theta. The erythrocyte sedimentation rate ESR(theta) = (d l/dt)t----0 determined from the above empirical equation increases with the increase in sin theta roughly linearly. The experimental results are discussed with reference to the Ponder-Nakamura-Kuroda theory and some recent theories.     

Steady flow through collapsible tubes: measurements of flow and geometry.

Compliant tubes attain a complex three-dimensional geometry when the external pressure exceeds the internal pressure and the tube is partially collapsed. A new technique for remote measurement of dynamic surfaces was applied to classical experiments with collapsible tubes. This work presents measurements of the three-dimensional structure of the tube as well as pressure and flow measurements during static loading and during steady-state fluid flow. Results are shown for two tubes of the same material and internal diameter but with different wall thicknesses. The measured tube laws compare well with previously published data and suggest the possible existence of a similarity tube law. The steady flow measurements did not compare well with the one-dimensional theoretical predictions.     

A semi-empirical model for flow of blood and other particulate suspensions through narrow tubes

A semi-empirical model applicable to the flow of blood and other particulate suspensions through narrow tubes has been developed. It envisages a central core of blood surrounded by a wall layer of reduced hematocrit. With the help of this model the wall layer thickness and extent of plug flow may be calculated using pressure drop, flow rate and hematocrit reduction data. It has been found from the available data in the literature that for a given sample of blood the extent of plug flow increases with decreasing tube diameter. Also for a flow through a given tube it increases with hematocrit. The wall layer thickness is found to decrease with increase in blood hematocrit. A comparison between the results of rigid particulate suspensions and blood reveals that the thicker wall layer and smaller plug flow radius in the case of blood may be attributed to the deformability of the erythrocytes.     

Pressure excursions during oscillatory flow in a branching network of tubes

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

Experimental analysis of the influence of stenotic geometry on steady flow.

We studied the flow behavior under steady flow conditions in four models of cylindrical stenoses at Reynolds numbers from 150 to 920. The flow upstream of the constrictions was always fully developed. The constriction ratios of the rigid tubes (D) to the stenoses (d) were d/D = 0.273; 0.505; 0.548; 0.786. The pressure drop at various locations in the stenotic models was measured with water manometers. The flow was visualized with a photoelasticity apparatus using an aqueous birefringent solution. We also studied the flow behavior at pulsatile flow in a dog aorta with a constriction of 71%. The flow through stenotic geometries depends on the Reynolds number of the flow generated in the tube and the constriction ratio d/D. At low d/D ratios, (with the increased constriction), the flow separation zones (recirculation zones, so-called reattachment length) and flow disturbances increased with larger Reynolds numbers. At lower values, eddies were generated. At high Re, eddies were observed in the pre-stenotic regions. The pressure drop is a function of the length and internal diameter of the stenosis, respective ratio of stenosis to the main vessel and the Reynolds numbers. At low Re-numbers and low d/D, distinct recirculation zones were found close to the stenosis. The flow is laminar in the distal areas. Further experiments under steady and unsteady flow conditions in a dog aorta model with a constriction of 71% showed similar effects. High velocity fluctuations downstream of the stenosis were found in the dog aorta. A videotape demonstrates these results.     

Numerical simulation of steady flow in a model of the aortic bifurcation.

The governing equations of steady flow of an incompressible viscous fluid through a 3-D model of the aortic bifurcation are solved with the finite element method. The effect of Reynolds number on the flow was studied for a range including the physiological values (200 < or = Re < or = 1600). The symmetrical bifurcation, with a branch angle of 70 degrees and an area ratio of 0.8, includes a tapered transition zone. Secondary flows induced by the tube curvature are observed in the daughter tubes. Transverse currents in the transition zone are generated by the combined effect of diverging and converging walls. Flow separation depends on both the Reynolds number and the inlet wall shear.     

Comparison of steady and pulsatile flow in a double branching arterial model

Data are presented to compare fluid flow parameters for steady flow with those for time-varying flow in a simplified two branch model which simulates the region of the abdominal aorta near the celiac and superior mesenteric branches of the dog. Measurements in the model included laser doppler anemometry velocity profiles during steady flow, sinusoidal flow with a superimposed mean flow (referred to as simple oscillatory flow) and arterial pulsatile flow. Shear rate measurements were made by an electrochemical technique during steady flow. Flow visualization studies were done during steady and pulsatile flow. Fluid flow effects in the simplified model during steady flow showed many similarities to the results from previous steady flow studies in a canine aortic cast. Shear rates in the region of the proximal (first, or celiac) branch were independent of flow rates in the distal (second, or mesenteric) branch, but the shear pattern within the proximal branch changed significantly as flow in the proximal branch increased. Shear rates on the proximal flow divider (leading edge into the distal branch) depended primarily on the flow rate to the proximal branch, but not on flow to the distal branch. At certain daughter branch flow ratios (approximately 2:1, proximal to distal), flow separation was promoted at the outer wall of the second branch, but flow separation did not occur in the first branch. In contrast to the canine aortic case results, flow separation was never detected on the distal (mesenteric) flow divider of the simplified model. This observation reflects the subtle effects of geometry on flow since the mesenteric flow divider in the canine cast protrudes into the main flow whereas the distal flow divider in the simplified model does not. There were distinct differences in the flow phenomena between steady, simple oscillatory and arterial pulsatile flow. Peak shear rates during pulsatile flow were as much as 10--100 times greater than steady flow shear rates at comparable mean flow rates. Particularly noteworthy for the pulsatile flow with a Womersley parameter of sixteen were very blunt velocity profiles throughout systole, and the absence of flow separation or reversal in those regions of the model that exhibited flow separation during steady flow. The shape of the waveform influences the nature of the flow during time-varying flows. Future studies of fluid dynamics in model systems must consider the pulsatile nature of the flow if a true interpretation of arterial flow phenomena is to be made.     

Blood flow downstream of a two-dimensional bifurcation

Many cardiovascular lesions such as aneurysms, intimal cushions, and atherosclerotic plaques tend to occur near bifurcations. This suggests that hemodynamic factors may be involved. Since measuring devices (such as anemometers) are still too large to allow local measurements of flow disturbances, we have attempted to predict the nature of these factors mathematically. Biological variables include pulsatile flow of a nonNewtonian fluid in distensible branching vessels with different angles and flow rates. Our initial analysis considers the flow in a two-dimensional bifurcation with a symmetrical flow divider perfused with steady flow at variable Reynolds numbers. At all flows, high shear forces develop on either side of the flow divider (i.e. at the apex of the bifurcation). With high flows, regions of sluggish or reverse flow develop near the outer walls of the bifurcation. The analysis confirms that the flow at the apex is quite different from that at the outer angles and that the latter varies more with flow rate than the former.     

A three-dimensional model of the upper tracheobronchial tree

A new model of the upper tracheobronchial tree is proposed to account for the three-dimensional nature of the airway system. In addition to the tube length, the tube diameter, and the branching angle, the model includes information on the orientation angle of each tube relative to its parent tube. The orientation angle, defined as the angle between two successive bifurcations, is useful for calculating the gravitational inclination of each tube. The information on orientation angle is further used to construct a binary coding system for identifying individual tubes in the airway tree. The proposed model is asymmetrical, but the same principles can be readily used to construct a symmetrical one.     

Simulation of coronary artery revascularization

Simulation of the commonly constructed geometries of aorto-coronary bypass anastomoses was carried out using especially fabricated distensible tubes and a pulsatile pump. The system pressure was maintained between 80 and 120 mmHg. The total mean flow was set at 250 ml min-1 (Reynolds number of 200) and the pulsatile frequency was varied from 0 to 2 Hz. A water-glycerine mixture having a density and viscosity similar to that of blood was used throughout. A 16 mm film of the front of black dye injected proximal to the anastomosis was made as the dye approached and passed through the anastomosis. Anastomotic geometries consisted of: end to side, parallel, 45 degree angle, and 90 degree angle. Stenoses, located in the tube representing the coronary artery, were simulated using a bevelled insert which represented an 80-85% area reduction. Flow visualization revealed that distensible tubes gave more realistic flow patterns than rigid tubes, a result particularly evident when a stenosis was present. Pulsatile flow demonstrated considerably more mixing than steady flow. The use of pulsatile flow in distensible tubing with a partial stenosis showed retrograde flow through the stenosis which was not evident for either steady flow or for flow in rigid tubing. The flow at the anastomatic site of the graft having an angle of 0 degrees showed a jetting action with a zone of recirculating fluid being present whereas for a 90 degree graft a distinct helical flow was formed distal to the anastomosis.     

Wall shear rate measurements in an elastic curved artery model

Since atherosclerotic lesions tend to be localized at bends and branching points, knowledge of wall shear rate patterns in models of these geometries may help elucidate the mechanism of atherogenesis. This study uses the photochromic method of flow visualization to determine both the mean and amplitude of the wall shear rate waveform in straight and curved elastic arterial models to demonstrate the effects of curvature, elasticity, and the phase angle between the flow and pressure waveforms (impedance phase angle). Under sinusoidal flow conditions characteristic of large arteries, the mean shear rate at the inner wall of the curved tube is reduced 40–56% from its steady flow value, depending on the phase angle. Wall shear rate amplitudes in the curved tube are significantly reduced by wall motion (36–55% of the Womersley amplitude for a straight rigid tube). The shear rate amplitude at the outer wall decreases 30% as the phase angle is reduced from −20° to −66°, while the shear rate amplitude at the inner wall increases 45%. As a result, the oscillatory nature of flow at the outer wall decreases with decreasing negative phase angle, but flow at the inner wall becomes much more oscillatory. At large negative phase angles, characteristic of hypertension or vasoactive agents, the shear rate at the inner wall has a small mean and cycles through positive and negative values; the shear rate at the outer wall remains positive throughout the flow cycle. Thus, the impedance phase angle could affect atherogenesis along the inner wall if temporal and directional changes in wall shear rate play a role.     

Erythrocyte sedimentation rate III tube diameter dependence in saline solution

Erythrocyte sedimentation rate was measured in a physiological saline solution as a function of both the tube diameter d and the initial suspension length iota 0. All the sedimentation curves in the vertical tubes were found to overlap over the range 1 mm less than d less than 7 mm and 100 mm less than iota 0 less than 330 mm, within the precision of 8 %. The sedimentation curves in the tilted tubes fit well to an exponential equation of iota = a [1 - exp (-bt)], where iota and t are the medium length along the tube and the elapsed time from sample injection respectively: At fixed tilt angle theta and iota 0, a was roughly constant and b was roughly proportional to l/d, while at fixed theta and d, a was linearly proportional to iota 0 and b was constant. The initial slope ESR (theta) = (d iota/dt) t----0 = ab was represented by a unique straight line as a function of iota 0/d for each fixed tilt angle. The experimental results were compared with some recent theories.     

The flow properties of axoplasm in a defined chemical environment: influence of anions and calcium.

The flow properties of axoplasm have been studied in a defined chemical environment. Axoplasm extruded from squid giant axons was introduced into porous cellulose acetate tubes of diameter roughly equal to that of the original axon. Passage of axoplasm along the tube rapidly coated the tube walls with a layer of protein. By measuring the rate of low back and forth along the tube, the rheological properties of the axoplasm plug were investigated at a range of pressures and in a variety of media. Axoplasm behaves as a classical Bingham body the motion of which can be characterized by a yield stress (theta) and a plastic viscosity (eta p). In a potassium methanesulphonate medium containing 65 nM free Ca2+, theta averaged 109 +/- 46 dyn/cm2 and eta p1 146 +/- 83 P. These values were little affected by ATP, COLCHICINE, CYTOCHOLASIN B or by replacing K by Na but were sensitive to the anion composition of the medium. The effectiveness of different anions at reducing theta and eta p1 was in the order SCN greater than I greater then Br greater than Cl greater than methanesulphonate. Theta and eta p1 were also drastically reduced by increasing the ionized Ca. This effect required millimolar amounts of Ca, was unaffected by the presence of ATP and was irreversible. It could be blocked by the protease inhibitor TLCK. E.p.r. measurements showed that within the matrix of the axoplasm gel there is a watery space that is largely unaffected by anions or calcium.     

Blood flow distribution in the branches of frog mesentery arterial microvessels]

The blood flow distribution in 49 arterial branchings of the mesentery (R. temporaria) was investigated (D of the trunk = 25.7 + 0.0 mum). Linear rate was measured by the impulse digital chronometry of the intervals of the erythrocyte transit time. The geometric characteristics of the branching was determined in vivo, on photographs. An asymmetric structure of the investigated branching was shown; branch 1 had the inner initial cross-section which was 2.2 times greater than that of branch 2 and lesser turning angles (29 and 59 degrees). The blood flow in branch 1 was three times greater than the blood flow in branch 2; this was due to its greater inner initial cross-section and a higher linear rate. According to calculations, the blood flow resistance of the branch-turn was insignificant in the general blood flow resistance of branches; therefore the turning angle of the branches could not serve as an important regulator of the volume of the blood flowing in them. An experimentally revealed association between the blood flow in the branches, their radius and their turning angles is well described by equations of the "optimal" model of the vessel branching.