Donald F. Young

Computer simulation of arterial flow with applications to arterial and aortic stenoses

A computer model for simulating pressure and flow propagation in the human arterial system is developed. The model is based on the one-dimensional flow equations and includes nonlinearities arising from geometry and material properties. Fifty-five arterial segments, representing the various major arteries, are combined to form the model of the arterial system. Particular attention is paid to the development of peripheral pressure and flow pulses under normal flow conditions and under conditions of arterial and aortic stenoses. Results show that the presence of severe arterial stenoses significantly affects the nature of the distal pressure and flow pulses. Aortic stenoses also have a profound effect on central and peripheral pressure pulse formation. Comparison with the published experimental data suggests that the model is capable of simulating arterial flow under normal flow conditions as well as conditions of stenotic obstructions in a satisfactory manner.

Pressure drop across artificially induced stenoses in the femoral arteries of dogs.

Stenoses were artificially induced in 13 large mongrel dogs by implanting small hollow cylindrical plugs in their femoral arteries. The instantaneous pressure drop across the stenosis and the flow rate were measured for a series of stenoses varying in severity from 52.3 to 92.2%. Mean pressure drops ranged from approximately 2 to 30 mm Hg with peak pressure drops ranging from 9 to 53 mm Hg. The pressure drop could be estimated from a relatively simple equation that was originally developed for flow through model stenoses. With this equation, the effects of several factors that contribute to the pressure drop, including stenosis size and shape, artery lumen diameter, blood density, blood viscosity, and velocity and acceleration of flow, could be clearly delineated. For severe stenoses, unsteady flow effects were small, and flow could be treated as quasi-steady. Calculations based on data obtained from the dog experiments revealed that the mean pressure drop across a stenosis increased nonlinearly with percent stenosis and showed quantitatively that the value of critical stenosis decreased with increasing demand for blood flow.

Flow through a converging-diverging tube and its implications in occlusive vascular disease — I

Abstract The steady flow of an incompressible fluid through an axisymmetric converging-diverging tube has been studied both theoretically (Part I) and experimentally (Part II). Possible implications of this type of flow in the development of vascular lesions are discussed in Part II. A mathematical model for a mild stenosis is established and an approximate solution for flow through a converging-diverging tube obtained. Velocity profiles, pressures, and wall shearing stresses along the tube are determined from this solution. The critical Reynolds number required for separation, and the extent of the separated region are also determined.

Hemodynamics of arterial stenoses at elevated flow rates.

This study is concerned with the pressure drop that develops across an arterial stenosis, with particular emphasis on the effect of the stenosis at high blood flow rates induced by a locally administered vasodilator drug. Stenoses, ranging in severity from 55.7% to 91.0% reduction in lumen area were artificially induced in the femoral and carotid arteries of large mongrel dogs. Instantaneous flow rates and pressure drops were measured over a wide range of flow conditions. Mean velocities varied from 3.9 to 88.8 cm/sec. Experimental data support the applicability of a relatively simple equation for predicting the pressure drop over this wide range of velocities and stenosis geometries. Results show that blood flow through a particular artery can increase by a large factor, in the range of 4-5, under conditions of vasodilation with a corresponding large decrease in pressure distal to the stenosis. The pressure drop increases in a nonlinear manner with velocity and thereby accentuates the importance of the stenosis at elevated flow rates. We suggest that a critical stenosis be defined in terms of its effect on maximal flow rather than resting flow.

Effect of geometry on pressure losses across models of arterial stenoses

Abstract The influence of geometric characteristics such as shape, length, and percent lumen area reduction on the pressure drop across an arterial stenosis is studied by means of an extensive series of in vitro steady-flow tests utilizing model constrictions in the form of blunt-ended hollow plugs. Results show that the pressure drop can be accurately predicted by a relatively simple equation. Extension of the results to shapes other than blunt plugs is also considered and shown to be possible. The pressure drop across multiple stenoses is investigated by considering two blunt plugs in series. Experimental data indicate that in general the pressure drop cannot be obtained by a symmation of pressure drops for single stenoses since the proximal and distal stenoses “interfere” with each other unless the spacing between them exceeds some critical distance which depends on the Reynolds number.

Initiation of turbulence in models of arterial stenoses

Abstract The development of turbulence under both steady and pulsatile flow through models of arterial stanoses was studied experimentally. Stenoses were represented by three severely constricted rigid-walled models with different shapes. Model geometries included a streamlined shape, a hollowed plug with blunt ends, and a thin plate orifice. Velocity fluctuations were measured with hot-film probes. Results indicate that: (a) turbulence develops at Reynolds numbers well below the critical value for flow in an unobstructed tube; (b) the critical Reynolds number varies with a dimensionless frequency parameter, first becoming less stable and then more stable as the frequency parameter is increased; and (c) the critical Reynolds number depends on the shape of the obstruction with the orifice-type stenosis exhibiting the lowest value for the critical Reynolds number.

WALL VIBRATIONS INDUCED BY FLOW THROUGH SIMULATED STENOSES IN MODELS AND ARTERIES

Abstract The paper describes a series of model and in vivo experiments in which vessel-wall vibrations. due to flow through simulated arterial stenoses, are investigated. The model studies utilize various axisymmetric constrictions inserted into two different flexible tubes. The flow through the models is steady, with the Reynolds numbers Re, ranging from 400 to 5000. Wall vibration intensity, I, a measure of the wall vibration amplitude, is found to be described by: I = K( D d ) 4 (Re) 2.2 where the coefficient K is dependent in part upon tube wall properties, D is the unobstructed vessel diameter, and d is the lumen diameter of the stenosis. Frequency spectra of wall vibrations and the axial position of maximum vibration intensity are also described. The in vivo studies utilize similar constrictions inserted into canine femoral arteries, and a qualitatively similar relation between wall-vibration intensity and fluid-dynamic parameters was observed. The results of this study correlate closely with the hydrodynamics of a separated, axisymmetric, bounded jet.

An in vitro study of flow response by cells.

Abstract Living cells grown as an attached, confluent monolayer are subjected to a steady, uniform laminar flow. The cells are grown in a specially designed cell culture chamber in which the flow conditions over the cells may be precisely defined and are experimentally verified. The flow conditions at the cell proximity are described, and the cellular and subcellular responses to the observed flow conditions are presented. The observed responses of the cells in the attached geometry are described for shear stresses ranging from 10 −3 to 10 1 dyn/cm 2 . The relative importance of the rheological properties of the cellular components is seen to vary with the magnitude of the shear stress applied. The significance of such flow studies is presented in relation to their possible use in evaulating mechanical site predilection in atherosclerosis.

FLOW THROUGH A CONVERGING-DIVERGING TUBE AND ITS IMPLICATIONS IN OCCLUSIVE VASCULAR DISEASE- II THEORETICAL AND EXPERIMENTAL RESULTS AND THEIR IMPLICATIONS*

Abstract In Part I of this communication, equations are developed to approximately describe the flow characteristics of an incompressible fluid through an axisymmetric converging-diverging tube. In Part II, an experimental program to study some of these flow characteristics is described, the theoretical and experimental results are compared, and some speculations are made on their implications in occlusive vascular disease. A plastic converging-diverging tube was constructed to determine the separation and reattachment points and the pressure drop across the entire stenosis. Water and blood were used in the experiments. Data were obtained for these fluids for Reynolds numbers up to approximately 1000. The experimental results obtained for the water and blood did not differ greatly, but in general only fair agreement was found between the experimental and theoretical results.

Effect of collateral and peripheral resistance on blood flow through arterial stenoses

Abstract A relatively simple analytical model is developed to investigate the interaction between blood flow through an arterial stenosis and the corresponding collateral and peripheral vascular beds. The model utilizes a realistic non-linear pressure-flow relationship for the stenosis. Two specific examples are studied in detail: the first relates to the hind limb of the dog, and the second to the human coronary system. Results indicate how normal resting flow can be maintained to the peripheral beds, even in the presence of a relatively severe stenosis, due to collateral flow and autoregulation of peripheral bed resistance. Whereas, under conditions of increased flow, the stenosis becomes the limiting element in the system and the vascular bed reserve is significantly reduced, even for a moderate stenosis. Predictions based on the model compare favorably with the limited in vivo data available. A discussion of the concept of the “critical stenosis” is included.