Analytical relationship between arterial input impedance and the three-element Windkessel series resistance

The recently proposed energy-balance method for estimating the series resistance of the three-element Windkessel model is reformulated in the frequency domain. New mathematical expressions are analytically derived, involving Fourier harmonics of pulsatile arterial pressure and flow. It is shown that the series resistance of the arterial three-element Windkessel model can be expressed as a weighted sum of the arterial input impedance moduli.     

Computing total arterial compliance of the arterial system from its input impedance

The total arterial compliance of the arterial system was computed from its input impedance by expressing the impedance in terms of its frequency-response vector diagram (f.r.v.) The f.r.v. plot of a 3-element windkessel subjected to random pacing follows, theoretically, a circular path. Since the windkessel model serves as a good approximation for the arterial system, we have used the simple properties of its f.r.v. plot to obtain the compliance, which is otherwise normally determined from the peripheral resistance and the time constant of the diastolic pressure decay. The arterial compliance can also be determined from the impulse response function of the arterial system. Data obtained from dog experiments during no intervention, aortic occlusion and during occlusion of both carotid arteries have been analysed.     

Dynamic model of the short-term regulation of arterial pressure in the cat

The purpose of this study was to characterise the dynamics of the short-term control of arterial pressure in the cat with the aid of a model consisting of a nonlinear negative-feedback control system. The arterial system was described by a three element windkessel model (peripheral resistance, R, aortic characteristic impedance, R_c, and total arterial compliance, C). The resistance regulation was represented by a second-order system with static gain G_R, a damping factor σ and an undamped natural frequency ω_n. The resistance gain, G_R, and the windkessel parameters were obtained from measurements of aortic and venous pressures and cardiac output in two steady states. The parameters σ and ω_n were estimated from mean pressure and mean flow during the transient from control to the new steady state. Pressure reductions averaged 10 per cent and resistance changes averaged 12 per cent. Average windkessel model parameters in the control condition were: C=(25·9±6·1) 10⁻⁶ g⁻¹ cm⁴ s², R_c=(2·51±0·53) 10³ g cm⁻⁴ s⁻¹, R=(40·9±9·8) 10³ g cm⁻⁴ s⁻¹. Average estimates of parameters of the resistance regulator were: G_R=(4·14±2·38) 10⁻³ min ml⁻¹, ω_n = 1·0 ± 1·0 rad s⁻¹, σ=0·41±0·19. A satisfactory fit was found between model predicted and measured pressure. The results suggest that the dynamic short-term control of pressure is underdamped and oscillatory. The amplitude of these oscillations is affected by arterial compliance, suggesting an interaction between the arterial system and short-term resistance regulation.     

Lumped model of terminal aortic impedance in the dog

The aim of this study was the formulation of a minimal lumped model of the aortic impedance as seen in the abdominal aorta just downstream of the origin of renal arteries. At this location simultaneous measurements of pressure and flow were taken in four anesthetized and open-chest dogs (weight, 30.9±5.8 kg) under basal, vasodilated (sodium nitroprusside) and vasoconstricted (methoxamine) conditions. Using these measurements we identified and compared three lumped models, A, B, and C, with decreasing complexity from A to C. The frequency response of these models was given the general form of peripheral resistance,R _P, multiplied by the ratio between (a) two zeros and two poles (model A); (b) two zeros and one pole (model B); and (c) one zero and one pole (model C).R _P was calculated as the ratio of mean pressure to mean flow. The other model parameters (time constants, damping factors, and natural frequencies) were estimated by minimizing the sum of squared differences between experimental and model generated pulsatile flows. After parameter estimation, the F-test was applied to compare the goodness of data fit obtained from the three models. Results of this test and the analysis of parameter estimation errors indicated that model B was preferable with respect to models A and C. The analysis of general model performance was followed by a consideration of alternative specific model structures that are physically realizable. With the aid of a determined model structure we evaluated the overall compliance of terminal aortic circulation under a variety of vascular states induced by injection of vasoactive agents.     

Dynamics of the short-term regulation of arterial pressure: Frequency dependence and role of arterial compliance

The purpose of this study was to investigate the role of the interaction between peripheral resistance regulation and arterial compliance in the overall short-term regulation of mean arterial pressure. A nonlinear model previously proposed was linearised about control values of mean arterial pressure and cardiac output so that it could be reformulated in terms of transfer functions. The resulting pressure to pressure open-loop transfer function H(s) consists of a complex conjugate pair of poles (pertaining to the resistance regulating system) and a real pole (pertaining to the arterial system, 1/(R₀ C), R₀=control resistance). Such a structure suggests an interaction between the resistance regulation and the arterial compliance (C). Quantitative evaluation of this interaction was obtained by estimating the model parameters during partial vena cava occlusions in four cats. Using these parameters the time response of the open-loop control system to a pressure step was found to be underdamped and oscillatory in all four cats (damping factor ξ ranged from 0·20 to 0·66), the amplitude of oscillations depending on the value of ξ and on the relative amplitude of the arterial time constant (compliance and peripheral resistance) with respect to the time constant 1/(ξω_n). Bode diagrams of H(jω) showed that the resonance peak due to the resistance regulating system may not be detectable, either because of the relatively high value of ξ or because it is masked by the pole of the arterial system.     

Implication of the systolic hump in systemic arterial pressure waves

The systolic hump in the aortic blood pressure wave is defined as the aorticresistance component proportional to the aortic blood flow superimposed on the windkessel component. An electrical analogue comprising a series resistance (aortic resistance) plus a resistance (peripheral resistance) and capacitance (aortic compliance) in parallel (i.e. windkessel component) is used for analysis. Curve fitting using the leastsquares method is performed on calculated and measured blood pressure waves from dogs under haemodynamical conditions induced by infusion of three drugs (noradrenaline, isoproterenol and acetylcholine). The curve fitting RMS (root mean square) errors are <3% for blood pressure waves and <30% for blood flow waves, with good agreement between measured and calculated blood flow waveforms. Infusion of noradrenaline and acetylcholine is found to induce a significant decrease and increase in the aortic resistance, respectively. Although only a small fraction of the blood pressure wave, the systolic hump has a marked effect on the systolic pressure waveform.     

Influence of pulse pressure variation on the results of local arterial compliance measurement: A computer simulation study

A computer simulation has been performed to study the influence of the pulse pressure variation on the arterial compliance readings in regard to different calculation techniques and arterial wall elastic properties. We applied the derivative- and amplitude-based (delta-based) calculation techniques to the model of the pressure vs. arterial lumen area relationship of different arteries. The simulated pulse pressure increase resulted in an essential reduction of the delta-based compliance in its near maximum region, and in an increase or no change outside this region. In the case of the relationship of a lower steepness the alterations were smaller.     

A nonlinear model of the arterial vessels within a limb segment

The relationship between the arterial blood pressure and the volume of the arteries within a segment of an extremity is nonlinear. The present paper shows how the flow and volume pulsations of the arteries within a limb segment can be simulated taking this property into account. An electrical model was constructed comprising one resistor and two voltage dependent ‘capacitors’, the latter corresponding to the pressure dependent elasticity, or compliance, of the arteries. Adequate simulations were obtained over a wide pressure range, which is impossible with linear models. The nonlinear, i.e. pressure dependent, relationship between the volume and pressure of arteries, observed under static conditions, must also be taken into consideration when studying pulsatile events with models whether mathematical or physical.     

Assessment of the influence of the compliant aortic root on aortic valve mechanics by means of a geometrical model

In recent years several researchers have suggested that the changes in the geometry and angular dimensions of the aortic root which occur during the cardiac cycle are functional to the optimisation of aortic valve function, both in terms of diminishing leaflet stresses and of fluid-dynamic behaviour. The paper presents an analytical parametric model of the aortic valve which includes the aortic root movement. The indexes used to evaluate the valve behaviour are: the circumferential membrane stress and the stress at the free edge of the leaflet, the index of bending strain, the bending of the leaflet at the line attachment in the radial and circumferential directions and the shape of the conduit formed by the leaflets during systole. In order to evaluate the role of geometric changes in valve performance, two control cases were considered, with different reference geometric configuration, where the movement of the aortic root was ignored. The results obtained appear consistent with physiological data, especially with regard to the late diastolic phase and the early ejection phase, and put in evidence the role of the aortic root movement in the improvement of valve behaviour.     

Identification of a fluorescein tracer model for determination of the flow rate of aqueous humor in the eye.

The continuous flow of the aqueous humor through the posterior and anterior chambers in the eye into Schlemm's canal is vital for the maintenance of a constant intraocular pressure. If the intraocular pressure rises above the normal range, several problems start appearing insidiously; for example, gradual visual field defects develop, leading ultimately to glaucoma. The determination of the flow rate of the aqueous humor is here performed by identifying a mathematical model after the free fluorescein concentration in the blood and in the anterior chamber of the eye has been measured. Experimental data obtained on rabbits are used in the analysis, and the flow rate of the aqueous humor in rabbit eyes is computed. This study demonstrates the feasibility of the approach in a clinical environment.