Physiological interpretation of inductance and low-resistance terms in four-element windkessel models: assessment by generalized sensitivity function analysis

Physiological relevance of parameters of three arterial models, denominated W4P, W4S and IVW, was assessed by computation of parameter-related generalized sensitivity functions (GSFs), which allow the definition of heart-cycle time intervals where the information content of experimental data, useful for estimation of each model parameter, is concentrated. The W4P and W4S are derived from the three-element windkessel by connecting an inductance, L, in parallel or in series, respectively, with aortic characteristic impedance, R(c). In the IVW, L is placed in series at the input of a viscoelastic windkessel, incorporating a Voigt cell (a resistor, R(d), in series with a capacitor, C). Pressure and flow measured in the ascending aorta of five ferrets and five dogs were used to estimate all model parameters, by fitting to pressure. For each model structure, parameter-related GSFs were generated. Focusing on controversial L, R(c) and R(d) physical meaning, our GSF analysis yielded the conclusion that, in both the W4S and the IVW, but not in the W4P, the L-term is suitable to represent the inertial properties of blood motion. Moreover, the meaning of aortic characteristic impedance ascribed to R(c) is questionable; while R(d) is likely to account for viscous losses of arterial wall motion.     

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.     

A comparative study of pulsatile arterial hemodynamics in rabbits and guinea pigs.

Pulsatile pressure and flow were measured in the ascending aorta and other arteries of 22 anesthetized rabbits and 16 anesthetized guinea pigs. Pressure/flow relationships were expressed as vascular impedance. Aortic flow waves were almost identical in the two species, but pressure waves were quite different. Reflected pressure waves returned earlier from the periphery in guinea pigs, augmenting pressure during late systole and resulting in relatively high external left ventricular work, an inappropriately larger difference between mean systolic and mean diastolic pressure and absence of any aortic diastolic pressure wave. Values of impedance modulus and phase were similar but differed in the frequency at which maxima and minima occurred. In both species, impedance curves were interpreted to indicate a functionally discrete reflecting site in the lower body whose position corresponded to the region of the aortic bifurcation. In addition, rabbits showed evidence of an upper body reflecting site approximately one-third as far distant from the heart. As in dogs, the arterial system in both species can be represented by an asymmetrical T-shaped model of realistic dimensions.     

Role of total arterial compliance and peripheral resistance in the determination of systolic and diastolic aortic pressure.

The goal of the study was to define the major arterial parameters that determine aortic systolic (Ps) and diastolic (Pd) pressure in the dog. Measured aortic flows were used as input to the two-element windkessel model of the arterial system, with peripheral resistance calculated as mean pressure over mean flow and total arterial compliance calculated from the decay time in diastole. The windkessel model yielded an aortic pressure wave from which we obtained the predicted systolic (Ps, wk) and diastolic (Pd, wk) pressure. These predicted pressures were compared with the measured systolic and diastolic pressures. The measurements and calculations were carried out in 7 dogs in control conditions, during aortic occlusion at four locations (the trifurcation, between trifurcation and diaphragm, the diaphragm and the proximal descending thoracic aorta) and during occlusion of both carotid arteries. Under all conditions studied the predicted systolic and diastolic pressure matched the experimental ones very well: Ps, wk = (1.000 +/- 0.0055) Ps with r = 0.958 and Pd, wk = (1.024 +/- 0.0035) Pd with r = 0.995. Linear regression for pulse pressure gave PPwk = (0.99 +/- 0.016) PP (r = 0.911). We found the accuracy of prediction equally good under control conditions and in presence of aortic or carotid artery occlusions. Multiple regression between pulse pressure and arterial resistance and total arterial compliance yielded a poor regression constant (r2 = 0.19) suggesting that the two arterial parameters alone cannot explain pulse pressure and that flow is an important determinant as well. We conclude that, for a given ejection pattern (aortic flow), two arterial parameters, total arterial resistance and total arterial compliance are sufficient to accurately describe systolic and diastolic aortic pressure.     

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.     

Identification of the three-element windkessel model incorporating a pressure-dependent compliance

A new one-step computational procedure is presented for estimating the parameters of the nonlinear three-element windkessel model of the arterial system incorporating a pressure-dependent compliance. The data required are pulsatile aortic pressure and flow. The basic assumptions are a steadystate periodic regime and a purely elastic compliant element. By stating two conditions, zero mean flow and zero mean power in the compliant element, peripheral and characteristic resistances are determined through simple closed form formulas as functions of mean values of the square of aortic pressure, the square of aortic flow, and the product of aortic pressure with aortic flow. The pressure across as well as the flow through the compliant element can be then obtained so allowing the calculation of volume variation and compliance as functions of pressure. The feasibility of this method is studied by applying it to both simulated and experimental data relative to different circulatory conditions and comparing the results with those obtained by an iterative parameter optimization algorithm and with the actual values when available. The conclusion is that the proposed method appears to be effective in identifying the three-element windkessel even in the case of nonlinear compliance.     

Central cardiovascular dynamics of ducks.

Blood pressures and flows have been recorded from the heart and arterial arches of ducks in order to present a complete picture of central hemodynamics in an avian species. An attempt has also been made to define the characteristics of the avian central circulation in terms of either "windkessel", or wave-transmission models. Mean arterial pressure (143 plus or minus 2.2 mmHg) and cardiac output (219 plus or minus 7ml/kg per min) were high compared with those of similarly sized mammals, although therewas no evidence of elevated pulmonary pressures, perhaps because of the unusual structure of the avian lung. Only 25% of the total systemic flow was distributed by the aorta, the remainder supplying the wing, flight muscles, and head via the brachiocephalic arteries. The contours of central pressure and flow waves ressembled those recorded inmammal except that significant circulation impedance modulus graphs indicated that resistance to pulsatile flow fell sharply to a minimum of 1/30th the DC value at 9-12 Hz. Flow led pressure at frequencies below 9-12Hz and pressure led flow at higherfrequencies. Impedance modulus in the pulmonary circulation fell to one-half the DC value and remained constant over a wide range of frequencies with pressure and flow being in a phase at all frequencies. Aortic pulse wave velocity varied with position inthe aorta as did the pressure pulse profile: these factors obviously limit the applicability of a windkessel model to the avian circulation.     

The central arterial pulses

Summary In order to examine the contours of central aortic and coronary flow pulses as well as those of pressure and flow pulses along the aorta, a hybrid model of the arterial system and the heart was designed. The digitally programmed model of the aortic system is an inhomogeneous transmission line with adjustable reflection factors at the end and at three intermediate locations. For the reflection factor at the entrance different values may be chosen for the ejection time and the diastole. The influence of a stenosis and of frequency-independent damping may be examined.The model of the ventricle is the analog solution of a system consisting of an internal isometric pressure source and an internal resistance and capacitance.The model of the coronary artery is the analog solution of a windkessel model of the system.The digital and analog models are interlocked by AD and DA converters. All programs can be executed in real time.The characteristic contour of the aortic flow is determined by the relation between the internal impedance of the ventricle and the magnitude of the characteristic impedance of the aorta. Furthermore, the influence of reflections within the arterial system is shown to be quite remarkable. Natural pressure pulses can be simulated by the model under normal and pathological conditions including aortic coarctation.The contour of the left coronary artery flow is determined on the one hand by the fraction of the ventricular pressure which acts as a counterpressure at the site of the peripheral resistance, and by the time constant of the coronary windkessel on the other hand.This work was supported by the Austrian research fund (Österreichischer Fonds zur Förderung der wissenschaftlichen Forschung).     

Computer identification of models for the arterial tree input impedance: Comparison between two new simple models and first experimental results

Two new simple models for the arterial tree input impedance are presented. The first is a 4-element windkessel model, obtained by paralleling the characteristic resistance of the 3-element windkessel model (Westerhof, 1968) with an inductance. The second is a tube model with a complex load, which is characterised by four parameters also. The parameters of these models are estimated from aortic root pressure and flow, which are either simulated by a complex numerical model of the peripheral arterial tree or measured in dogs. The parameter estimation is carried out using an automatic procedure based on the Powell optimisation algorithm. The importance of the evaluation of the achievable accuracy in the estimation of parameters and its relationship to model identifiability is emphasised.     

Assessment of distributed arterial network models

The aim of this study is to evaluate the relative importance of elastic non-linearities, viscoelasticity and resistance vessel modelling on arterial pressure and flow wave contours computed with distributed arterial network models. The computational results of a non-linear (time-domain) and a linear (frequency-domain) mode were compared using the same geometrical configuration and identical upstream and downstream boundary conditions and mechanical properties. Pressures were computed at the ascending aorta, brachial and femoral artery. In spite of the identical problem definition, computational differences were found in input impedance modulus (max. 15–20%), systolic pressure (max. 5%) and pulse pressure (max. 10%). For the brachial artery, the ratio of pulse pressure to aortic pulse pressure was practically identical for both models (3%), whereas for the femoral artery higher values are found for the linear model (+10%). The aortic/brachial pressure transfer function indicates that pressure harmonic amplification is somewhat higher in the linear model for frequencies lower than 6 Hz while the opposite is true for higher frequencies. These computational disparities were attributed to conceptual model differences, such as the treatment of geometric tapering, rather than to elastic or convective non-linearities. Compared to the effect of viscoelasticity, the discrepancy between the linear and non-linear model is of the same importance. At peripheral locations, the correct representation of terminal impedance outweights the computational differences between the linear and non-linear models.     

Comparative value of respiratory input and transfer impedances in field studies

Total respiratory impedance was obtained from 4 to 30 Hz in 39 healthy males and in 140 iron miners by studying the relationship between transrespiratory pressure and mouth flow when forced oscillations were applied to the respiratory system in two different ways: by varying pressure at the mouth (input impedance, Zin); by varying pressure around the chest (transfer impedance, Ztr). Zin was characterized by the slope (S) and intercept (R0) of the resistance-frequency curve, by the resonant frequency (fn), and by total respiratory inertance (I) and compliance (C) obtained from the reactance. Transfer resistance and reactance curves were respectively analysed with a two-parameter (m1, m3) and a three-parameter (m0, m2, m4) equation. Significant correlations were found between impedance indices, particularly between m1 and both R0 (r = 0.931; p less than 0.001) and m2 (r = 0.739; p less than 0.001), and also between impedance indices and maximal expiratory flows. Nonsmoking miners (n = 40), compared to control nonsmokers (n = 16), had slightly lower maximal flows at high lung volumes (p less than 0.05) and higher values of m1 (p less than 0.001), R0, S, m3 (p less than 0.01) and m2 (p less than 0.05). In contrast, miners who smoked (n = 46) mainly differed from miners who did not smoke by lower flows at middle and low lung volumes (p less than 0.01) and larger m0 (p less than 0.01), m2 and m4 (p less than 0.05).(ABSTRACT TRUNCATED AT 250 WORDS)     

硝普钠对动物血管力学参数的影响

有研究表明硝普钠具有增加动脉顺应性的趋势和降低外周阻力的作用,但是硝普钠对血管中血流惯性的影响却不统一。本文以Coldwyn等建立的动脉系改良风箱力学模型为主要研究方法对动物动脉血管顺应性(包括中央和外周顺应性),血流惯性和外周阻力等进行了研究,并建立了动脉系总阻抗的公式。然后评价硝普钠对总阻抗的影响以及总阻抗做为一种血管力学参数的灵敏性。实验结果表明:硝普钠具有统计学意义地增加外周血管的顺应性(P<0.05);有增加中央动脉顺应性和血流惯性的趋性;并能降低外周阻力和动脉系总阻抗(P<0.05)。结果还表明动脉系总阻抗是一种灵敏度高的血管力学参数。因此,硝普钠能影响血管参数,对外周血管顺应性和血管总阻抗的作用尤其明显。     

Assessment of aortic pressure power components and their link to overall elastic and resistive arterial properties

This paper reviews the analytical expressions for in-phase and quadrature aortic power components associated with the real and imaginary parts of aortic input admittance, respectively. It is shown that active power Wact, and its steady, Wstdy, and pulsatile, Wpuls, components logically follow from in-phase power. Reactive power follows from quadrature power only for sinusoidal signals. The definition of reactive power indexes for real aortic pressures and flows requires extreme care. The link between overall arterial properties and pressure power components (and indexes) is investigated, making use of a three-element windkessel model and ascending aortic pressure and flow data taken from eight anaesthetised dogs, under basal state and after treatment with a vasoconstrictor (methoxamine). Seven dogs are normotensive in the baseline state (NBA cases, n = 7), the average (+/- SE) of mean pressure being 86.5 +/- 5.2 mmHg. The eighth dog has a baseline mean pressure of 134 mmHg and is considered to be hypertensive. The two experimental cases from this dog are grouped with those from the other seven dogs after vasoconstriction, to form the NVC + H group (n = 9). On average, fitting the model to the experimental data yields a 100% increase (p < 0.05) in total peripheral resistance, a 63% decrease (p < 0.01) in total arterial compliance and a 10% decrease (p > 0.05) in aortic characteristic impedance, from the NBA group to the NVC + H. Correspondingly, the peak-to-peak amplitude of quadrature power shows a 69% increase (p < 0.02). Wact, Wstdy, and Wpuls show a 28% increase (p > 0.05), a 40% increase (p < 0.02) and a 43% decrease (p > 0.05), respectively. Energetic efficiency of the arterial system, Eart = 1 - (Wpuls/Wact), increases by 8% (p < 0.02). From analysis of the estimates of power components and arterial parameters in relation to low-frequency phase angles of aortic impedance, it is concluded that the decrease in total arterial compliance with increasing pressure reduces the power lost in pulsation. This happens at the expense of an increase in quadrature power and absolute values of related reactive power indexes.     

左心循环系统的建模与仿真

将左心模型与四元件的动脉系统Windkessel模型耦合,构成左心-动脉系统交互的左心循环系统模型.模型包括左心房、左心室、二尖瓣、动脉辩和动脉系统,实现了对左心循环系统的血流动力学模拟.应用状态空间法和SIMULINK框图模型法两种技术和MATLAB工具,进行了数学建模和数值计算,具有模型直观、容易实现、方便调节参数等优点.应用这一仿真模型,可以对左心室容积、血压及主动脉血压和血流等进行动态模拟.仿真结果与生理实际情况相符.     

A sensitivity analysis of a personalized pulse wave propagation model for arteriovenous fistula surgery. Part A: Identification of most influential model parameters

Previously, a pulse wave propagation model was developed that has potential in supporting decision-making in arteriovenous fistula (AVF) surgery for hemodialysis. To adapt the wave propagation model to personalized conditions, patient-specific input parameters should be available. In clinics, the number of measurable input parameters is limited which results in sparse datasets. In addition, patient data are compromised with uncertainty. These uncertain and incomplete input datasets will result in model output uncertainties. By means of a sensitivity analysis the propagation of input uncertainties into output uncertainty can be studied which can give directions for input measurement improvement. In this study, a computational framework has been developed to perform such a sensitivity analysis with a variance-based method and Monte Carlo simulations. The framework was used to determine the influential parameters of our pulse wave propagation model applied to AVF surgery, with respect to parameter prioritization and parameter fixing. With this we were able to determine the model parameters that have the largest influence on the predicted mean brachial flow and systolic radial artery pressure after AVF surgery. Of all 73 parameters 51 could be fixed within their measurement uncertainty interval without significantly influencing the output, while 16 parameters importantly influence the output uncertainty. Measurement accuracy improvement should thus focus on these 16 influential parameters. The most rewarding are measurement improvements of the following parameters: the mean aortic flow, the aortic windkessel resistance, the parameters associated with the smallest arterial or venous diameters of the AVF in- and outflow tract and the radial artery windkessel compliance.     

Can a linear electrical analog model of a mechanical valve predict flow by using a pressure gradient?

The objective was to determine whether a previously developed technique for biological aortic valves could predict flow through a mechanical valve. An electrical analog model of the aortic valve that includes compliance, resistance, and inertance parameters, and corresponding second order differential equations was used to predict flow given a pressure gradient, as previously reported. Simulated pressures and flow were recorded by using a pulse duplicator system. The heart rate was varied from 60 to 180 bpm, and the stroke volume was varied from 22 to 67 cc. Resistance, inertance, and compliance parameters of the governing differential equation were estimated by using a least-squares fit to the measured flow at 120 bpm and 50 cc stroke volume. By using these parameter estimates, flow was calculated for other heart rates and stroke volumes. To achieve a better flow prediction, a nonlinear filter (third order polynomial range calibration equation) was applied to the output of the linear model (flow). The mean error, full-scale error, and spectral error in magnitude and phase between measured and predicted flow were compared. Error in mean flow ranged from 3% at medium flow rates to 90% at low flow rates. The maximum and minimum full scale errors were 12% and 5%, respectively. Error in the harmonics of measured and calculated flow ranged from 0% to 55%. Larger errors were usually present at the higher harmonics. The agreement between measured and calculated flow was better at normal and high flows but rather poor at low flows. The nonlinear filter (range calibration equation) was unable to account for the discrepancies between the measured and calculated flow over all flow ranges. It seems that this linear model and nonlinear filter have limited application, and an alternate nonlinear approach may produce better results.     

Simple and accurate way for estimating total and segmental arterial compliance: The pulse pressure method

We derived and tested a new, simple, and accurate method to estimate the compliance of the entire arterial tree and parts thereof. The method requires the measurements of pressure and flow and is based on fitting the pulse pressure (systolic minus diastolic pressure) predicted by the two-element windkessel model to the measured pulse pressure. We show that the two-element windkessel model accurately describes the modulus of the input impedance at low harmonics (0–4th) of the heart rate so that the gross features of the arterial pressure wave, including pulse pressure, are accounted for. The method was tested using a distributed nonlinear model of the human systemic arterial tree. Pressure and flow were calculated in the ascending aorta, thoracic aorta, common carotid, and iliac artery. In a linear version of the systemic model the estimated compliance was within 1% of the compliance at the first three locations. In the iliac artery an error of 7% was found. In a nonlinear version, we compared the estimates of compliance with the average compliance over the cardiac cycle and the compliance at the mean working pressure. At the first three locations we found the estimated and “actual” compliance to be within 12% of each other. In the iliac artery the error was larger. We also investigated an increase and decrease in heart rate, a decrease in wall elasticity and exercise conditions. In all cases the estimated total arterial compliance was within 10% of mean compliance. Thus, the errors result mainly from the nonlinearity of the arterial system. Segmental compliance can be obtained by subtraction of compliance determined at two locations.     

Model analysis of the contribution of atrial contraction to ventricular filling

We studied the contribution of atrial contraction to ventricular filling by modelling the right heart and the associated vasculature. Right atrial and ventricular contractions were represented by periodically varying volume elastances which are independent of loading conditions. The values of these elastances were experimentally determined. The systemic veins, the tricuspid valve and the pulmonary arteries were all represented by impedance networks. For these impedances we used as much experimentally obtained information as possible. The dynamic pressure and flow waveforms observed in the model under control conditions generally agreed with those reported in the literature. We therefore proceeded to analyze the effects of changing the time interval between atrial systole and ventricular systole, atrial contractility, heart rate, and blood inertance. There was an optimal atrial systole-ventricular systole interval of about 0.1 sec for ventricular filling. Stroke volume of the ventricle was enhanced by an increase in atrial contractility, a decrease in heart rate, and an increase in blood inertance. The effect of changing atrial compliance was found to be dependent on heart rate. Contribution of atrial contraction to ventricular filling was also found to be more significant during exercise than at rest.     

Left ventricular output derived from the time-derivative and phase velocities of the aortic pressure wave

A method of deriving aortic flow from pressures measured in the proximal aorta has been tested. The Womersley equation was used with the pressure gradient substituted by the time derivative of the pressure and the measured phase velocity c′ of each harmonic component. Reflected waves alter the value of c′ and are a source of error when a constant value of c is used, as in previous methods in which dP/dt has been used. The predicted mean flow was compared with that measured with a flowmeter. The results were N=720, r=0·97, y=1·00x+0·19 ml/s, s.e.e. ±12%. This was not significantly altered by a series of interventions which produced a wide range in stroke outputs. The prediction of a pulsatile flow wave was less accurate than that for mean flow. Analysis of the results shows that this is due to the fact that any method using the time derivative of the pressure cannot predict the phase of the input impedance; this leads to a distortion of the waveform on resynthesis. The amplitude of the impedance is accurately measured by the present method even when reflections markedly alter the wave velocity, as at slow heartrates. The results for predicting mean flow are shown to be better than those reported by other methods.     

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.