Abraham Noordergraaf

Analog studies of the human systemic arterial tree

Abstract The design, construction and evaluation of a linear passive model of the human systemic arterial tree are presented. The performance of this electrical model is compared extensively to its real counterpart in the following areas: magnitude and phase of input impedances, wave travel (amplitude and phase of pressure harmonics) along the aorta, and wave shapes of pressures and flows at different locations. These comparisons demonstrate that the model behaves very much like the real system. A series of refinements in the modeling of a short segment of artery is discussed; although these refinements help to obtain better agreement with reality, none has a major effect on the behavior of the model as measured close to the heart. Reflections play a major role in determining the behavior of the system and occur at all branch points. The largest reflection coefficients are found at the periphery. These reflection coefficients result mainly from the architecture of the arterial tree. It is shown how the nature of the input impedance and wave travel pattern can be explained in terms of these reflections. The input impedance and wave travel in large vessels, for frequencies larger than 2 cps, are largely determined by the characteristics of these vessels themselves and not by the load. This is mainly due to the architecture of the system. Alterations in peripheral resistance affect the input impedance of the system only for very low frequencies; the same holds for wave travel in the aorta: the high frequencies are virtually independent of the peripheral resistance. Some clinical states are simulated and discussed.

Arterial tonometry: Review and analysis

A review is presented of the field of arterial tonometry and of the problems involved with its application. A second generation model is developed which interprets most of the difficulties encountered in previous experimental work. The model also identifies barriers that must be overcome to allow tonometry to become a practical technique for obtaining measurement of continuous, absolute blood pressure. Problems addressed include those of calibration, positioning sensitivity, design standardization, material properties and vascular loading characteristics. Theoretical and experimental studies provide support for the application of basic biomechanical concepts for solution of these problems and suggest required design features.

Arterial viscoelasticity: A generalized model: Effect on input impedance and wave travel in the systematic tree

Abstract In the first part of this paper most of the reported results concerning measurements of the viscoelastic properties of the systemic arterial wall are discussed. The various mechanical models of the vessel wall that have been proposed and which usually account for a special aspect of viscoelastic properties are reviewed critically. From these discussions a new mathematical model for the wall properties emerges. It accounts in quantitative terms for the frequency dependence of the Young modulus, stress-relaxation, creep, and hysteresis. Hence this new description, which is in terms of the complex Young modulus, covers all the known aspects of the viscoelastic wall properties. In the second part of this paper the complex Young modulus is incorporated in a model of the systematic arterial tree for the purpose of studying the effect of the viscous properties of the wall. Wave travel and input impedances are given for the case of a purely elastic wall and for the case of a realistic viscoelastic wall; the differences are compared. Addition of the viscous wall properties proves to have a significant effect on input impedance and wave travel.

Pulse wave propagation.

This report evaluates pulse wave propagation with respect to contributions by vascular wall elastic and geometric properties, vessel wall and blood viscosity, and noniinearities in system parameters and in the equations of motion. Discrepancies in results obtained with different experimental methods and theory are discussed and resolved. A three-point pressure technique was used to obtain measurements from the abdominal aorta, carotid, iliac, and femoral arteries of dogs. Computations involved linear, as well as nonlinear methods. Results are presented along a continuous path of transmission (abdominal aorta, iliac, femoral), and it is shown that variations in phase velocity can be explained entirely by the geometric variation of these vessels. Phase velocities are shown to be frequency independent at 54 Hz whereas attenuation increases progressively for higher frequencies. Determination of propagation coefficients using maximal, compounded values of reported viscoelastic and geometric properties just manages to span the range of phase velocities, determined in different laboratories, but does not do so for attenuation. Also, differences in experimental techniques cannot explain these discrepancies. Consideration of geometric taper, nonlinear compliance, all the terms in the equation of motion, and the effect of wall and blood viscosity resolves discrepancies betweeen theoretical models and experimentally derived phenomena.

Reduced Models of Arterial Systems

Simple models that manifest input impedances of arterial systems are compared. An improvement upon documented two-, three-, and five-element models is presented. The classical two-element model (the windkessel) accounts for the lowest frequency components, and the three-element model (the modified windkessel) accounts for both low-and high-frequency components of the spectrum of interest. Five-element models, however, by allowing for reflection, can account for principal features over the entire frequency range of interest.

Thresholds for premature ventricular contractions in frog hearts exposed to lithotripter fields

Piezoelectrically generated lithotripter shocks were shown to produce premature ventricular contractions of the frog heart. Anesthetized grass frogs, Rana pipiens, were studied following implantation of an aortic catheter and EKG leads. The most sensitive phase of the heart cycle for the generation of premature ventricular contractions with lithotripter shocks at 30 MPa peak pressure was found to be the T-P segment. During this phase of the heart cycle, the minimum peak-positive pressure shock wave necessary to produce a premature ventricular contraction in a frog heart was between 5 MPa and 10 MPa.

Right ventricular-pulmonary arterial interactions

The application of pulsatile models to hemodynamic data has made possible a more complete understanding of the relationship of pulmonary pressure and flow. To review the genesis of these concepts, the unique characteristics of the pulmonary artery and right ventricle are outlined as a basis for understanding why differences in their pulsatile properties from the systemic circuit must exist. The pulmonary impedance spectrum is introduced and the concept of optimal right ventricular-pulmonary artery coupling is explored based on a review of extensive experimental data. Finally, available studies of normal pulmonary impedance in man and abnormal impedance in human disease states are reviewed, with emphasis on disturbances in optimal ventricular-vascular coupling. The important implications of these concepts for understanding and treatment of cardiovascular disease are developed.

The Korotkoff sound

As the auscultatory method of blood pressure measurement relies fundamentally on the generation of the Korotkoff sound, identification of the responsible mechanisms has been of interest ever since the introduction of the method, around the turn of the century. In this article, a theory is proposed that identifies the cause of sound generation with the nonlinear properties of the pressure-flow relationship in, and of the volume compliance of the collapsible segment of brachial artery under the cuff. The rising portion of a normal incoming brachial pressure pulse is distorted due to these characteristics, and energy contained in the normal pulse is shifted to the audible range. The pressure transient produced is transmitted to the skin surface and stethoscope through deflection of the arterial wall. A mathematical model is formulated to represent the structures involved and to computer the Korotkoff sound. The model is able to predict quantitatively a range of features of the Korotkoff sound reported in the literature. Several earlier theories are summarized and evaluated.

Input Impedance, Wave Travel, and Reflections in the Human Pulmonary Arterial Tree: Studies Using an Electrical Analog

An electrical model of the human pulmonary arterial tree has been developed using modern oscillatory flow theory. By comparison with corresponding in vivo measurements of 1) pressure and flow waveshapes, 2) input impedance, and 3) apparent phase velocity, this passive, multisegmental, linear delay line model is shown to bear a close resemblance to reality.

Cross-Sectional Shape of Collapsible Tubes

In order to quantify the collapse phenomenon in veins, this paper presents a mathematical analysis of the cross-sectional shape of a flexible tube as its internal pressure varies. Quantitative results are presented in terms of the physical parameters of the tube, such as wall thickness and Youngs modulus. It is assumed that the tube is thin walled, that no stretching occurs, that the cross-sectional shape is elliptical when the transmural pressure is zero, and that the longitudinal prestress is zero. The equations were solved on a digital computer which displayed the cross-sectional shapes on an oscilloscope, which were then photographed. A selection of these photographs is presented. Curves are shown which give the cross-sectional area and compliance as functions of transmural pressure. Other curves are shown which are useful for interpolation, and for use in the experimental determination of the physical parameters which may otherwise be difficult or impossible to measure accurately.