Reduced Models of Arterial Systems
01 Feb 1985-IEEE Transactions on Biomedical Engineering (IEEE Trans Biomed Eng)-Vol. 32, Iss: 2, pp 174-176
TL;DR: Simple models that manifest input impedances of arterial systems are compared and an improvement upon documented two-, three-, and five-element models is presented.
Abstract: 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.
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TL;DR: In chronically treated heart failure patients with full angiotensin-converting enzyme-inhibition and diuretics, there is decreased compliance of the aorta and its major branches, which is inversely correlated with the aldosterone escape phenomenon.
Abstract: Aims The purpose of this study was to examine, in chronically treated heart failure patients vs control subjects, the influence of neurohumoral activation and aldosterone escape on arterial elastic behaviour, assessed by noninvasive mathematical lumped-parameter modelling of the compliance of the arterial system.
Methods and Results Radial arterial pulse waves were recorded non-invasively for 30 s with an arterial tonometer sensor array in 13 chronic heart failure patients (mean age, 59 +/- 2.5 years) in New York Heart Association class II. The patients had been taking digoxin, furosemide, captopril and aspirin for more than 3 months. Thirteen healthy subjects (mean age, 50 +/- 4.0 years) acted as controls. Compliance of the proximal (aorta and major branches, C1) and distal parts (C2) of the circulation were derived from a third order four-element modified Windkessel model which can reproduce arterial pressure waveforms, including both exponential and oscillatory sections. Active renin, angiotensin II and aldosterone levels were determined on venous blood samples in the supine position and after 30 min active standing. There was decreased proximal (C1, 1.51 +/- 0.11 ml . mmHg(-1), P<0.01) and distal (C2, 0050+/-0.011 ml . mmHg(-1)) arterial compliance in the chronic heart failure patients vs controls (C1, 1.71 +/- 0.16 ml. mmHg(-1); C2, 0.054 +/- 0.006 ml . mmHg(-1)). The chronic heart failure patients were characterized by an aldosterone escape phenomenon which was inversely correlated with the proximal arterial compliance in both supine (r= -0.795, P=0.002) and standing (r= -0.628, P=0.029) positions.
Conclusions In chronically treated heart failure patients with full angiotensin-converting enzyme-inhibition and diuretics, there is decreased compliance of the aorta and its major branches, which is inversely correlated with the aldosterone escape phenomenon.
191 citations
TL;DR: A genuinely heterogeneous multiscale approach where the local model and the systemic one are coupled at a mathematical and numerical level and solved together and have no longer boundary conditions on the artificial sections, but interface conditions matching the two submodels.
Abstract: In this paper we propose a method for coupling distributed and lumped models for the blood circulation. Lumped parameter models, based on an analogy between the circulatory system and an electric or a hydraulic network are widely employed in the literature to investigate different systemic responses in physiologic and pathologic situations (see e.g. [13, 24, 30, 15, 4, 27, 11, 14]). From the mathematical viewpoint these models are represented by ordinary differential equations. On the other hand, for the accurate description of local phenomena, the Navier–Stokes equations for incompressible fluids are considered. In the multiscale perspective, lumped models have been adopted (see e.g. [16]) as a numerical preprocessor to provide a quantitative estimate of the boundary conditions at the interfaces. However, the two solvers (i.e. the lumped and the distributed one) have been used separately. In the present work, we introduce a genuinely heterogeneous multiscale approach where the local model and the systemic one are coupled at a mathematical and numerical level and solved together. In this perspective, we have no longer boundary conditions on the artificial sections, but interface conditions matching the two submodels. The mathematical model and its numerical approximation are carefully addressed and several test cases are considered.
151 citations
TL;DR: This paper starts with the one-dimensional axisymmetric Navier-Stokes equations for time-dependent blood flow in a rigid vessel to derive lumped models relating flow and pressure and focuses on zeroth, first and second order models and relate them to electrical circuit analogs, in which current is equivalent to flow and voltage to pressure.
Abstract: Windkessel and similar lumped models are often used to represent blood
flow and pressure in systemic arteries. The windkessel model was originally developed by Stephen Hales (1733) and Otto Frank (1899) who used it to describe blood flow in the heart. In this paper we start with the one-dimensional axisymmetric Navier-Stokes equations for time-dependent blood flow in a rigid vessel to derive lumped models relating flow and pressure. This is done through Laplace transform and its inversion via residue theory. Upon keeping contributions from one, two, or more residues, we derive lumped models of successively higher order. We focus on zeroth, first and second order models and relate them to electrical circuit analogs, in which current is equivalent to flow and voltage to pressure. By incorporating e ffects of compliance through addition of capacitors, windkessel and related lumped models are obtained. Our results show that given the radius of a blood vessel, it is possible to determine the order of the model that would be appropriate for analyzing the flow and pressure in that vessel. For instance, in small rigid vessels ($R <$ 0.2 cm) it is adequate to use Poiseuille's law to express the relation between flow and pressure, whereas for large vessels it might be necessary to incorporate spatial dependence by using a one-dimensional model accounting for axial variations.
133 citations
Cites methods from "Reduced Models of Arterial Systems"
...In addition to the two resistors and the capacitor of the three-element windkessel model, the four-element model includes an inductor representing the inertia of the blood [16, 29, 31 ]....
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TL;DR: A detailed discussion of these issues forms the basis of this review and indicates that impaired pulsatile function of arteries provides important prognostic and therapeutic information beyond that provided by traditional blood pressure measurements.
Abstract: Ageing and disease states associated with an increase in cardiovascular events alter the physical characteristics of blood vessel walls and impair the pulsatile function of arteries. An accumulating body of evidence indicates that impaired pulsatile function of arteries provides important prognostic and therapeutic information beyond that provided by traditional blood pressure measurements. A variety of techniques are currently employed to evaluate the mechanical properties of arteries. All techniques have theoretical, technical and practical limitations that impact on their widespread application in the clinical setting and use as measurement tools to improve cardiovascular risk stratification. A detailed discussion of these issues forms the basis of this review.
133 citations
TL;DR: A windkessel model with two resistors and a capacitor to reproduce beat-to-beat changes in middle cerebral artery blood flow velocity in response to arterial pressure changes measured in the finger suggests that the initial increase in cerebrovascular resistance can explain the widening of the cerebral blood flow pulse observed in young subjects.
Abstract: The dynamic cerebral blood flow response to sudden hypotension during posture change is poorly understood. To better understand the cardiovascular response to hypotension, we used a windkessel model with two resistors and a capacitor to reproduce beat-to-beat changes in middle cerebral artery blood flow velocity (transcranial Doppler measurements) in response to arterial pressure changes measured in the finger (Finapres). The resistors represent lumped systemic and peripheral resistances in the cerebral vasculature, whereas the capacitor represents a lumped systemic compliance. Ten healthy young subjects were studied during posture change from sitting to standing. Dynamic variations of the peripheral and systemic resistances were extracted from the data on a beat-to-beat basis. The model shows an initial increase, followed approximately 10 s later by a decline in cerebrovascular resistance. The model also suggests that the initial increase in cerebrovascular resistance can explain the widening of the cerebral blood flow pulse observed in young subjects. This biphasic change in cerebrovascular resistance is consistent with an initial vasoconstriction, followed by cerebral autoregulatory vasodilation.
129 citations
References
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TL;DR: Blood pressure and velocity waveforms were recorded in a series of patients at cardiac catheterization and the changes in shape of the waveforms are interpreted in terms of reflections and are related to computations of vascular input impedance.
Abstract: Blood pressure and velocity waveforms were recorded in a series of patients at cardiac catheterization. The changes in shape of the waveforms are interpreted in terms of reflections and are related to computations of vascular input impedance.
344 citations
TL;DR: The results suggest that the subjects of group C were better able to meet the increased energy demands imposed by an abnormally high aortic input impedance, and whether the high impedances in subjects with coronary disease represent an increase with age and transmural pressure alone, or whether some additional factor is involved.
Abstract: To determine the systemic input impedance, pulsatile pressure and flow were measured in the ascending aorta in 16 human subjects who were undergoing diagnostic cardiac catheterization. Blood flow was measured with a catheter-tip electromagnetic velocity meter, and pressure with an external transducer connected with the fluid-filled lumen of the catheter. Five subjects were found to have no evidence of cardiovascular disease (group A, mean age 32 +/- 2 years, mean aortic pressure 97 +/- 4 mm Hg). Seven had clinical and angiographic signs of coronary arterial disease, and mean pressures less than 100 mm Hg (group B, mean age 48 +/- 2 years). Four subjects had signs of coronary disease and mean pressures greater than 100 mm Hg (group C, mean age 48 +/- 3 years). The frequency spectra of impedance were qualitatively similar in all three groups and resembled those previously observed in the canine aorta. Characteristic impedance was lower in the normal subjects (group A, average 53 dyn sec cm-5) than in the subjects with coronary artery disease (groups B and C, average 129 dyn sec cm-5). Among the subjects with coronary disease, characteristic impedance was higher in the hypertensive subjects (group C, average 202 dyn sec cm-5) than in those with lower mean pressures (group B, average 95 dyn sec cm-5). External left ventricular work per unit time (hydraulic power) averaged 1715 milliwatts (mW) in group A, 1120 mW in group B, and 2372 mW in group C. Cardiac outputs were within normal limits in all subjects, but tended to be lower in group B than in group C. These results suggest that the subjects of group C were better able to meet the increased energy demands imposed by an abnormally high aortic input impedance. Further investigation is needed to learn whether the high impedances in subjects with coronary disease represent an increase with age and transmural pressure alone, or whether some additional factor is involved. The data on relatively normal subjects permit a tentative definition of the normal limits for aortic input impedance in man: 26-80 dyn sec cm-5.
303 citations
280 citations
TL;DR: The elasticity of the pulmonary arterial tree appears to be as important as the state of the arterioles and capillaries in determining the energy required for pulsatile pulmonary blood flow.
Abstract: The differential pressure method of Womersley and McDonald was used to measure instantaneous blood flow in the main pulmonary artery in ten human subjects. Three subjects had normal pulmonary arterial pressures and flows, seven had mitral stenosis and pulmonary hypertension. The spectrum of input impedance versus frequency was similar to that previously reported for the dog and rabbit, with the modulus decreasing from relatively high values at zero frequency to a minimum between 2 and 5 cycles/sec. Characteristic impedance and phase velocity were lower in the normal subjects than in those with pulmonary hypertension (averages, 23 dyne sec cm -5 and 1.68 m/sec in the normals; 46 dyne sec cm -5 and 4.77 m/sec in the hypertensives). Hydraulic energy dissipated per unit time by pulsations in the pulmonary bed was usually higher in the hypertensive than in the normal cases, because of the greater stiffness of the pulmonary arteries in the subjects with pulmonary hypertension. The elasticity of the pulmonary arterial tree appears to be as important as the state of the arterioles and capillaries in determining the energy required for pulsatile pulmonary blood flow.
233 citations