Apparent stresses in disturbed pulsatile flows

Traditional attempts at decomposing measured velocities into repeatable and random components are examined for a set of velocity data measured under pulsatile flow conditions distal to a 90% axisymmetric constriction. The Reynolds numbers, which are typical of those found in the human carotid artery, are such that transitional phenomena occur during portions of the pulsatile cycle at several axial stations. The implications of the method selected for velocity decomposition upon the computation of fluctuating or 'apparent' stresses is a point of major focus. It is shown that the usual estimation of Reynolds stresses in a pulsatile flow by subtracting the ensemble-averaged velocity from the instantaneous velocity leads to an underestimation of the apparent stress when coherent or repeatable disturbances exist in the flow. An alternative decomposition using a frequency domain approach is presented which combines both random and coherent stresses into a single apparent stress, and it is proposed that this approach is preferable to the traditional ensemble averaging method when estimating fluctuating stresses in arterial flows.     

Pulsatile poststenotic flow studies with laser Doppler anemometry

The pulsatile flow field distal to axisymmetric constrictions in a straight tube was studied using laser Doppler anemometry. The upstream centerline velocity waveform was sinusoidal at a frequency parameter of 7.5 and mean Reynolds number of 600. Stenosis models of 25, 50 and 75% area reduction were employed and velocity data were derived by ensemble averaging methods. Extensive measurements of the pulsatile velocity profiles are reported, and wall shear rates were computed from the near wall velocity profile gradients. The experiments indicate that a permanent region of poststenotic flow separation does not exist even for the severest constriction, in contrast to results for steady flow. Values of wall shear stress were greatest near the throat of the constriction and were relatively low in the poststenotic region, including the region of most intense flow disturbance. Turbulence was found only for the 75% stenosis model and was created only during a segment of the cycle. Although much emphasis has been placed upon turbulence in the detection of arterial stenoses, particularly as identified by Doppler ultrasound spectral broadening, the present study implies that identification of flow disturbances of an organized nature may be more fundamental in recognizing mild to moderate disease. Additionally, the relationship of these flow field results to the animal aortic coarctation model often employed in atherogenesis studies is discussed.     

The effect of asymmetry in abdominal aortic aneurysms under physiologically realistic pulsatile flow conditions

In the abdominal segment of the human aorta under a patient's average resting conditions, pulsatile blood flow exhibits complex laminar patterns with secondary flows induced by adjacent branches and irregular vessel geometries. The flow dynamics becomes more complex when there is a pathological condition that causes changes in the normal structural composition of the vessel wall, for example, in the presence of an aneurysm. This work examines the hemodynamics of pulsatile blood flow in hypothetical three-dimensional models of abdominal aortic aneurysms (AAAs). Numerical predictions of blood flow patterns and hemodynamic stresses in AAAs are performed in single-aneurysm, asymmetric, rigid wall models using the finite element method. We characterize pulsatile flow dynamics in AAAs for average resting conditions by means of identifying regions of disturbed flow and quantifying the disturbance by evaluating flow-induced stresses at the aneurysm wall, specifically wall pressure and wall shear stress. Physiologically realistic abdominal aortic blood flow is simulated under pulsatile conditions for the range of time-average Reynolds numbers 50 < or = Rem < or = 300, corresponding to a range of peak Reynolds numbers 262.5 < or = Repeak < or = 1575. The vortex dynamics induced by pulsatile flow in AAAs is depicted by a sequence of four different flow phases in one period of the cardiac pulse. Peak wall shear stress and peak wall pressure are reported as a function of the time-average Reynolds number and aneurysm asymmetry. The effect of asymmetry in hypothetically shaped AAAs is to increase the maximum wall shear stress at peak flow and to induce the appearance of secondary flows in late diastole.     

Disorder distal to modeled stenoses in steady and pulsatile flow

Measurements of the velocity and energy spectra were made in the distal region of modeled stenoses in a rigid tube with both steady and pulsatile water flows. Reynolds numbers of 318–2540 and a pulsatile flow frequency parameter of 15 were employed. The effects of the degree of stenosis, the stenosis geometry and the presence or absence of the downstream confining wall on the development of flow disturbances were investigated. Visualization of the distal flow patterns in stenotic and free jets illustrated the existence of complex fields which included vortex shedding, highly turbulent regions, and recirculation zones. Significant flow disorder was created by a mild stenosis in pulsatile, but not in steady, flow. Nondimensionalization employing the stenosis diameter and flow velocity in the throat of the constriction correlates the vortex shedding frequency and energy spectra within a limited postestenotic region.     

Flow-induced wall shear stress in abdominal aortic aneurysms: Part II--pulsatile flow hemodynamics

In continuing the investigation of AAA hemodynamics, unsteady flow-induced stresses are presented for pulsatile blood flow through the double-aneurysm model described in Part I. Physiologically realistic aortic blood flow is simulated under pulsatile conditions for the range of time-average Reynolds numbers 50< or =Re(m) < or =300. Hemodynamic disturbance is evaluated for a modified set of indicator functions which include wall pressure (p(w)), wall shear stress (tau(w)), Wall Shear Stress Gradient (WSSG), time-average wall shear stress (tau(w)*), and time-average Wall Shear Stress Gradient WSSG*. At peak flow, the highest shear stress and WSSG levels are obtained at the distal end of both aneurysms, in a pattern similar to that of steady flow. The maximum values of wall shear stresses and wall shear stress gradients are evaluated as a function of the time-average Reynolds number resulting in a fourth order polynomial correlation. A comparison between numerical predictions for steady and pulsatile flow is presented, illustrating the importance of considering time-dependent flow for the evaluation of hemodynamic indicators.     

Flow-induced Wall Shear Stress in Abdominal Aortic Aneurysms: Part II - Pulsatile Flow Hemodynamics

In continuing the investigation of AAA hemodynamics, unsteady flow-induced stresses are presented for pulsatile blood flow through the double-aneurysm model described in Part I. Physiologically realistic aortic blood flow is simulated under pulsatile conditions for the range of time-average Reynolds numbers 50 h Re m h 300. Hemodynamic disturbance is evaluated for a modified set of indicator functions which include wall pressure ( p w ), wall shear stress ( w ), Wall Shear Stress Gradient (WSSG), time-average wall shear stress ( w *), and time-average Wall Shear Stress Gradient WSSG *. At peak flow, the highest shear stress and WSSG levels are obtained at the distal end of both aneurysms, in a pattern similar to that of steady flow. The maximum values of wall shear stresses and wall shear stress gradients are evaluated as a function of the time-average Reynolds number resulting in a fourth order polynomial correlation. A comparison between numerical predictions for steady and pulsatile flow is presented, illustrating the importance of considering time-dependent flow for the evaluation of hemodynamic indicators.     

Blood flow in abdominal aortic aneurysms: pulsatile flow hemodynamics

Numerical predictions of blood flow patterns and hemodynamic stresses in Abdominal Aortic Aneurysms (AAAs) are performed in a two-aneurysm, axisymmetric, rigid wall model using the spectral element method. Physiologically realistic aortic blood flow is simulated under pulsatile conditions for the range of time-averaged Reynolds numbers 50< or =Re(m)< or =300, corresponding to a range of peak Reynolds numbers 262.5< or =Re(peak) < or = 1575. The vortex dynamics induced by pulsatile flow in AAAs is characterized by a sequence of five different flow phases in one period of the flow cycle. Hemodynamic disturbance is evaluated for a modified set of indicator functions, which include wall pressure (p(w)), wall shear stress (tau(w)), and Wall Shear Stress Gradient (WSSG). At peak flow, the highest shear stress and WSSG levels are obtained downstream of both aneurysms, in a pattern similar to that of steady flow. Maximum values of wall shear stresses and wall shear stress gradients obtained at peak flow are evaluated as a function of the time-average Reynolds number resulting in a fourth order polynomial correlation. A comparison between predictions for steady and pulsatile flow is presented, illustrating the importance of considering time-dependent flow for the evaluation of hemodynamic indicators.     

In vitro flow dynamics of four prosthetic aortic valves: a comparative analysis

The velocity fields downstream of four prosthetic heart valves were mapped in vitro over the entire cross-section of a model aortic root using laser Doppler anemometry. THe Bj?rk-Shiley 60 degrees convexo-concave tilting disc valve, the Smeloff-Cutter caged ball valve, the St. Jude Medical bileaflet valve, and the Ionescu-Shiley standard bioprosthesis were examined under both steady and pulsatile flows. Velocity profiles under steady flow conditions were a good approximation for pulsatile profiles only during midsystole. The pulsatile flow characteristics of the four valves showed variation in large scale flow structures. Comparison of the valves according to pressure drop, shear stress and maximum velocities are also provided.     

Numerical analysis of flow through a severely stenotic carotid artery bifurcation

The results of computational simulations may supplement MR and other in vivo diagnostic techniques to provide an accurate picture of the hemodynamics in particular vessels, which may help demonstrate the risks of embolism or plaque rupture posed by particular plaque deposits. In this study, a model based on an endarterectomy specimen of the plaque in a carotid bifurcation was examined. The flow conditions include steady flow at Reynolds numbers of 300, 600, and 900 as well as unsteady pulsatile flow. Both dynamic pressure and wall shear stress are very high, with shear values up to 70 N/m2, proximal to the stenosis throat in the internal carotid artery, and both vary significantly through the flow cycle. The wall shear stress gradient is also strong along the throat. Vortex shedding is observed downstream of the most severe occlusion. Two turbulence models, the Chien and Goldberg varieties of k-epsilon, are tested and evaluated for their relevance in this geometry. The Chien model better captures phenomena such as vortex shedding. The flow distal to stenosis is likely transitional, so a model that captures both laminar and turbulent behavior is needed.     

On the discrimination between band-limited coherent and random apparent stresses in transitional pulsatile flow

A frequency domain approach that incorporates a matched filter was examined for discriminating between ordered velocity fluctuations with band-limited frequency content and random velocity variations in pulsatile disturbed flows. Fluctuations at pseudo-discrete frequencies may yield a significant contribution to the apparent stress tensor computed from the unsteady Navier Stokes equations, and an estimate of the stresses arising from these ordered structures can be obtained once the velocity variations have been decomposed. This type of decomposition permits the estimation of the apparent stresses in turbulent flows, consisting of coherent and random parts, in blood flow applications such as diseased constricted arteries or downstream of artificial heart valves.     

Experimental study of pulsatile and steady flow through a smooth tube and an atherosclerotic coronary artery casting of man

In vitro investigation of pulsatile and steady flows through a smooth, straight circular tube and a diseased human coronary artery cast was conducted with sugar-water solutions simulating the viscosity of blood. Time averaged pressure drops for pulsatile flows measured in the circular tube over a Reynolds number ranging from 50 to 1,000 were found to be identical to those for steady flows in the same tube, both of which were in excellent agreement with the Poiseuille flow prediction. For the polyurethane case (# 124) made from a human main coronary with significant but 'non obstructive' diffuse atherosclerotic disease, pressure drops for steady flows were found to be greater than Poiseuille flow predictions by a factor of 3-8 in the physiological Reynolds number range from about 100 to 400. Pulsatile flows in the same artery cast resulted in additional 30% increases in time averaged pressure drops, and thus flow resistance, compared to the steady flow data. Steady and pulsatile flow data measured in a straight, axisymmetric model of cast # 124 showed considerably smaller increases in flow resistance than those observed in # 124 casting.     

Attenuation of flow disturbances in tapered arterial grafts

Flow disturbances in tapered arterial grafts of angles of taper between 0.5 and 1.0 deg were measured in vitro using a pulsed ultrasound Doppler velocimeter. The increase in transition Reynolds numbers with angle of taper and axial distance was determined for steady flow. The instantaneous centerline velocities were measured distal to a 50 percent area stenosis (as a model of a proximal anastomosis), in steady and pulsatile flow, from which the disturbance intensities were calculated. A significant reduction in post-stenotic disturbance intensity was recorded in the tapered grafts, relative to a conventional cylindrical graft. In pulsatile flow with a large backflow component, however, there was an increase in disturbance intensity due to diverging flow during flow reversal. This was observed only in the 1.0 deg tapered graft. These findings indicate that taper is an important consideration in the design of vascular prostheses.     

Nonquasi-steady character of pulsatile flow in human coronary arteries

This experiment was conducted to determine if the pulsatile flow through the proximal portion of the left coronary artery system in man exhibits quasi-steady characteristics. Steady and pulsatile flows were passed through an idealized model whose dimensions were based on a vascular cast. The mean Reynolds number was 180 and the unsteadiness number was 2.7. Velocity profiles were measured by laser Doppler anemometry at several locations along diameters in the parent and both daughter channels in the neighborhood of the "left main" bifurcation. Analysis of the results along one diameter in the "left main" channel shows that unsteady flow in the larger coronary arteries may not be simulated by a series of steady flow experiments.     

Pulsatile pressure and flow in the skeletal muscle microcirculation

Although blood flow in the microcirculation of the rat skeletal muscle has negligible inertia forces with very low Reynolds number and Womersley parameter, time-dependent pressure and flow variations can be observed. Such phenomena include, for example, arterial flow overshoot following a step arterial pressure, a gradual arterial pressure reduction for a step flow, or hysteresis between pressure and flow when a pulsatile pressure is applied. Arterial and venous flows do not follow the same time course during such transients. A theoretical analysis is presented for these phenomena using a microvessel with distensible viscoelastic walls and purely viscous flow subject to time variant arterial pressures. The results indicate that the vessel distensibility plays an important role in such time-dependent microvascular flow and the effects are of central physiological importance during normal muscle perfusion. In-vivo whole organ pressure-flow data in the dilated rat gracilis muscle agree in the time course with the theoretical predictions. Hemodynamic impedances of the skeletal muscle microcirculation are investigated for small arterial and venous pressure amplitudes superimposed on an initial steady flow and pressure drop along the vessel.     

Application of large-eddy simulation to the study of pulsatile flow in a modeled arterial stenosis

The technique of large-eddy simulation (LES) has been applied to the study of pulsatile flow through a modeled arterial stenosis. A simple stenosis model has been used that consists of a one-sided 50 percent semicircular constriction in a planar channel. The inlet volume flux is varied sinusoidally in time in a manner similar to the laminar flow simulations of Tutty (1992). LES is used to compute flow at a peak Reynolds number of 2000 and a Strouhal number of 0.024. At this Reynolds number, the flow downstream of the stenosis transitions to turbulence and exhibits all the classic features of post-stenotic flow as described by Khalifa and Giddens (1981) and Lieber and Giddens (1990). These include the periodic shedding of shear layer vortices and transition to turbulence downstream of the stenosis. Computed frequency spectra indicate that the vortex shedding occurs at a distinct high frequency, and the potential implication of this for noninvasive diagnosis of arterial stenoses is discussed. A variety of statistics have been also extracted and a number of other physical features of the flow are described in order to demonstrate the usefulness of LES for the study of post-stenotic flows.     

Numerical modeling of pulsatile turbulent flow in stenotic vessels

Pulsatile turbulent flow in stenotic vessels has been numerically modeled using the Reynolds-averaged Navier-Stokes equation approach. The commercially available computational fluid dynamics code (CFD), FLUENT, has been used for these studies. Two different experiments were modeled involving pulsatile flow through axisymmetric stenoses. Four different turbulence models were employed to study their influence on the results. It was found that the low Reynolds number k-omega turbulence model was in much better agreement with previous experimental measurements than both the low and high Reynolds number versions of the RNG (renormalization-group theory) k-epsilon turbulence model and the standard k-epsilon model, with regard to predicting the mean flow distal to the stenosis including aspects of the vortex shedding process and the turbulent flow field. All models predicted a wall shear stress peak at the throat of the stenosis with minimum values observed distal to the stenosis where flow separation occurred.     

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.